Question

finish the triangle proof by dragging the correct reasons to their box. there will be two extra unused reasons!
given: ab || de, be bisects ad
prove: ab ≅ de
be bisects ad
ab || de
△abc ≅ △dec
ab ≅ de
∠acb ≅ ∠dce
∠bac ≅ ∠edc
ac ≅ cd
reflexive property
cpctc
sas
alternate interior angles
given
given
asa

Explanation:

Step1: Reason for BE bisects AD

The reason for the first statement (BE bisects AD) is "Given" as it is provided in the problem statement.

Step2: Statement for Definition of Bisect

If BE bisects AD, then by the definition of bisect, \(AC = CD\).

Step3: Reason for AB || DE

The reason for the third statement (AB || DE) is "Given" as it is provided in the problem statement.

Step4: Angles due to parallel lines

Since \(AB\parallel DE\), by the Alternate - Interior Angles Theorem, \(\angle BAC=\angle EDC\).

Step5: Congruence of triangles

For \(\triangle ABC\) and \(\triangle DEC\), we have \(AC = CD\) (from step 2), \(\angle BAC=\angle EDC\) (from step 4), and \(\angle ACB=\angle DCE\) (Vertical Angles). So, by the ASA (Angle - Side - Angle) congruence criterion, \(\triangle ABC\cong\triangle DEC\).

Step6: Reason for \(AB = DE\)

Since \(\triangle ABC\cong\triangle DEC\), by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \(AB = DE\).

Answer:

1. Given
2. \(AC = CD\)
3. Given
4. \(\angle BAC=\angle EDC\); Alternate Interior Angles
5. ASA
6. CPCTC